Question regarding definition of $\mathbb{P}(A||\mathscr{G})$

Question

I'm currently study conditional probaility form the book Probability and measure Book by Patrick Billingsley and I stuck to understanding the following thing.

We know that $\mathbb{P}(A||\mathscr{G})$ is a random variable with two properties:

(1) $\mathbb{P}(A||\mathscr{G})$ is measurable $\mathscr{G}$ and integrable.

(2) $\mathbb{P}(A||\mathscr{G})$ satisfies the functional equation $$\int_G \mathbb{P}(A||\mathscr{G})dP=\mathbb{P}(A\cap G), \ \ \ G\in \mathscr{G}.$$

Condition (i) in the definition above in effect requires that the values of $\mathbb{P}(A||\mathscr{G})$ depends only on the sets in $\mathscr{G}$. An observer who knows the outcome of $\mathscr{G}$ viewed as an experiment knows for each $G$ in $\mathscr{G}$ whether it contains $\omega$ or not; for each $x$ he knows this in particular for the set $[\omega' :\mathbb{P}(A||\mathscr{G})_{\omega^{'}}=x]$, and hence he in principle knows the functional value $\mathbb{P}(A||\mathscr{G})_{\omega}$ even if does know $\omega$ itself.

I want a explanation and a simple example of this bold font statement.

Thanks in advance

Answer 1

The conditional probability $\mathbb{P}(A\mid \mid \mathscr{G})$ can be interpreted as the belief about the occurence of the event $A$ is one knows which events in the sigma-field $\mathscr{G}$ occur and which ones do not occur. The simplest and most meaningful scenario is when $\mathscr{G}$ is generated by a real-valued random variable $X$, that is the sigma-field made by all the events which can be represented as $\{\omega: X(\omega)\in B\}$ for some $B$ in the Borel sigma-field. Knowing which events in such sigma-field occur is equivalent to know the value of $X$. In this scenario, what is said in the bold statement is this: if one knows the value of $X$ then he/she also knows the conditional probability $\mathbb{P}(A\mid \mid \mathscr{G})$ even if he/she does not know $\omega$.

Answer 2

Condition (1) says that $g_A(\omega)=\mathbb{P}[A|\mathscr{G}][\omega)$ is a $\mathscr{G}$ measurable random variable, it can only be explain through events occurring in $\mathscr{G}$. In geometric terms, $g_A$ is he best approximation to the function $\mathbb{1}_A$ in terms of functions that are $\mathscr{G}$-measurable.

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Consider $(\Omega,\mathscr{F},\mathbb{P})=([0,1],\mathscr{B}([0,1]),\lambda)$ to be the unit interval with the Borel $\sigma$--finite algebra and Lebesgue measure.

Consider $\mathscr{G}=\sigma([0,1/2],(1/2,1])$. Let $A=(1/4,3/4]$ then

$g_A=P[A|\sigma(\mathscr{G})]=c_1\mathbb{1}_{[0,1/2]}+c_2\mathbb{1}_{(1/2,1]}$

where $c_1$ and $c_2$ are some constants. I leave it to you to find those constants. The point is that $g_A$ explains the event $A$ only with the information that is contained in $\mathscr{G}$.

If $\omega\in [0,1/2]$ then $g_A(\omega)=c_1$; if $\omega\in(1/2,1]$ , then $g_A(\omega)=c_2$